According to Shannon formula C=W·log21+S/N ), where C is the channel capacity, W is the channel bandwidth, S is the signal power and N is the noise power, it can be seen that the channel capacity is proportional to the bandwidth, therefore the most effective method to improve the channel capacity is to increase the bandwidth. In addition, it can be seen that the channel capacity can be also improved by increasing the signal power.
In a current communication system, different information is carried on different frequency bands using carrier modulation technologies to be transmitted on the frequency bands, the essence of which is to fully utilize bandwidth resources to improve the channel capacity. FIG. 1 shows a current typical carrier modulation principle. The real part of a baseband complex signal is multiplied by the carrier cos(ωt), and the imaginary part of the baseband complex signal is multiplied by the carrier sin(ωt), then the multiplied real part and the multiplied imaginary part are accumulated and transmitted finally. This process can be expressed by the following formula: SBP(t)=Re{SLP(t)eiωt}, where SBP(t) is the carrier modulation signal, eiωt is the complex carrier signal, SLP(t) is the baseband complex signal and Re represents taking the real part. The principle of this formula is the multiplication of time domain signals equals to the convolution of frequency domain signals. Baseband signals are shifted to carrier frequency bands via the convolution of carrier frequency signals and the baseband signals. Obviously, in the current carrier modulation methods, although the baseband signal is a complex number and the carrier signal is also a complex number, only the real part taken from the carrier modulation signals is transmitted finally. Therefore, real signals are transmitted, which is called real carrier modulation herein.
Actually, the current real carrier modulation methods have resulted in the multiplied waste of frequency spectrum resources and multiplied loss of signal energy as negative frequencies are not proper understood and used.
Firstly, negative frequencies do exist. As shown in FIG. 2, an angle of counterclockwise rotation is defined as +θ, and an angle of clockwise rotation is defined as −θ, then it can be learned based on the definition
  ω  =            ⅆ      θ              ⅆ      t      of an angular frequency that a negative angular frequency
      -    ω    =            ⅆ              (                  -          θ                )                    ⅆ      t      is generated by “negative angle” instead of “negative time”. Therefore, as a matter of fact, the positive and negative frequencies only represent that there are rotations in two different directions on a plane. In essence, the positive and negative rotations exist because the plane has two surfaces. The positive frequencies whose rotation directions accord with the right-hand rule are defined as right rotation frequencies herein, which are called right frequencies for short. The negative frequencies whose rotation directions accord with the left-hand rule are defined as left rotation frequencies herein, which are called left frequencies for short. Unless otherwise referred to, the positive and negative frequencies, the positive and negative frequency bands, and the positive and negative spectrums etc. in the existing technologies are replaced by terms such as left and right frequencies, left and right frequency bands, and left and right spectrums etc. hereinafter.
So far, no matter in teaching materials or in engineering implementation, the defined available bandwidths (also known as operating frequency bands) are within the range of right spectrums with positive signs, while left spectrums are abandoned selectively because of the negative signs in the mathematical expressions. FIG. 3 shows division of frequency bands in the most cutting-edge Long Term Evolution (LTE) communication protocol currently, and frequency spectrum resources with negative signs are completely neglected.
While understanding the natural existence of left frequencies, how to distinguish the left and right frequencies, or how to describe these two rotations on a plane? Euler's formula will give the answer: e±iωt=cos(ωt)±i sin(ωt). As shown in FIG. 4, e−iωt and eiωt represent a clockwise rotation curve and a counterclockwise rotation curve respectively, corresponding to the left and right frequency signals. Although the left and right frequency signals are easily distinguished in a “time-complex number” space, the projections of the left and right frequency signals are all real signals cos(ωt) apparently on a “time-real part” plane, i.e. Re{e−iωt}=Re{e+iωt}=cos(ωt). Therefore, when a real signal appears, it cannot be distinguished whether the signal is the projection of a left frequency signal or the projection of a right frequency signal; speaking from the probability, the signal is equally likely to be a left frequency signal or a right frequency signal, i.e. both the probability of being a left frequency signal and the probability of being a right frequency signal are ½, i.e. cos(ωt)=(e−iωte+iωt)/2. Therefore, real signals with only one degree of freedom are incomplete. A complex signal with at least two degrees of freedom is required to describe a frequency signal unambiguously. In order words, complete description of a frequency signal must be in a form of a complex number. In the complete description, the left and right frequencies e−iωt and eiωt in the form of the complex number are two completely independent frequencies which can be distinguished. Therefore, the left and right frequencies are able to carry completely independent information.
As analyzed above, real signals generated by real carrier modulation actually cause confusion of left and right frequencies, thus the left and right frequency bands are both occupied, and information on the left and right frequency bands are in conjugate symmetry and not independent. FIG. 5 shows a spectrum shifting in real carrier modulation, wherein the abscissa is the frequency ω, the ordinate is the amplitude F(ω) and ωC is the carrier frequency. By the way, in the real carrier modulation mode, since two-dimensional complex signals are observed from incomplete one-dimensional real signals, the left frequency band generated by the real carrier modulation mode has brought great confusion, and may be mistakenly assumed to be only a mirror image which does not really exist in case of not knowing the meaning and function of the left frequency band. A more serious point of view regards signals of the left frequency band harmful, thus bringing about many methods such as “mirror image inhibition” etc.
Currently, the received signals are regarded as real signals during demodulation, therefore multiplication, i.e. frequency band shifting is performed for real signals only. Generally, the right frequency band is shifted to a baseband. In this way, the left frequency band is shifted to a position, the distance from the position to the baseband is twice the distance from the left frequency band to the baseband before shifting, and all information of the left frequency band filtered by the baseband is erased. Although the mirror image information of the left frequency band is redundant, energy loss of the signal will be multiplied actually if the mirror image information of the left frequency band is abandoned directly. FIG. 6 shows a spectrum shifting in real carrier demodulation, wherein the abscissa is the frequency ω, the ordinate is the amplitude F(ω) and ωC is the carrier frequency. FIG. 7 shows energy loss in a process from transmitting signals to receiving the signals. A complete complex signal is a left-rotation or right-rotation plane signal (a). Having undergone the grating effect (b) of real carrier modulation, and the projection effect (c) of a receiving antenna, the energy loss of the actually received signal may be quadrupled or more. Luckily, such incomplete real carrier demodulation is applied because the information carried in the left and right frequencies is conjugate mirror information, thus it is the same to receive the information on the left frequency even if a demodulation terminal is confused with the left and right frequencies; in this case, it only needs to exchange the I data and Q data to mirror the information back, that's why many instruments are provided with an option for performing I, Q exchange for received signals.
It can be seen from the frequency band shifting process in the modulation and demodulation above that the frequency is actually a relative value which changes with the change of a reference frequency. The reference frequency here refers to a modulation and demodulation frequency and only the distance between the frequencies, i.e. the frequency band has an absolute meaning, which proves the actual existence of “negative frequencies” from another perspective.
To sum up, because of the natural bias to the left frequency, all bandwidth definitions included in all current communication systems including wireless, wire, optical fiber, radar and the like, neglect the frequency spectrum resources of the left frequency, which leads to a waste of half of the frequency spectrum resources. In addition, the left and right frequency bands are occupied in the current real carrier modulation, and either the left frequency band signal energy or the right frequency band signal energy is abandoned in the current real carrier demodulation.